First name is the given Name. In writing the full name when surname exists, in the states like Telangana and Andhra pradesh, the surname will be written first, followed by Given Name and middle name. This is usually the traditional British/English way of writing the surname at the beginning.
The Last name is usually the surname. It is referred as the family name. This remain constant in the entire family. In the states like Tamilnadu, the surname is the father name and written next to the given name. In the states of southern India, the Husband name could be added to the given name / maiden name which is typically again referred as the last name hence forth.
Full name is the complete name as it has to be read orderly including surname, first name, middle name and last name. Usually, the order of the full name is ~Given Name~ or ~First Name~, ~Middle Name~, ~Last name~ or ~Surname~
To change one's given name or name including the surname, one has to publish such change and intimate through public news papers. The person who would like to change their name need to give the notification in leading local news papers. Upon the notification, need to approach the respective gazette publication authority with respective forms filled and fee payable. You can consult your legal authority for more information.